Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2004.10606 · Journal of Statistical Mechanics Theory and Experiment

Scaling of the formation probabilities and universal boundary entropies in the quantum XY spin chain

Filiberto Ares, M. A. Rajabpour, Jacopo Viti

Published Aug 1, 2020• 7 citations
arXivHTMLPDF

Abstract

Abstract We calculate exactly the probability to find the ground state of the XY chain in a given spin configuration in the transverse σ z -basis. By determining finite-volume corrections to the probabilities for a wide variety of configurations, we obtain the universal boundary entropy at the critical point. The latter is a benchmark of the underlying boundary conformal field theory characterizing each quantum state. To determine the scaling of the probabilities, we prove a theorem that expresses, in a factorized form, the eigenvalues of a sub-matrix of a circulant matrix as functions of the eigenvalues of the original matrix. Finally, the boundary entropies are computed by exploiting a generalization of the Euler–MacLaurin formula to non-differentiable functions. It is shown that, in some cases, the spin configuration can flow to a linear superposition of Cardy states. Our methods and tools are rather generic and can be applied to all the periodic quantum chains which map to free-fermionic Hamiltonians.

DOI
10.1088/1742-5468/aba9d4

Related papers

Formation probabilities and Shannon information and their time evolution after quantum quench in the transverse-field XY chain

Khadijeh Najafi, M. A. Rajabpour

We first provide a formula to calculate the probability of occurrence of different configurations (formation probabilities) in a generic free fermion system. We then study the scaling of these probabilities with respect to the size in the case of the critical transverse-field XY chain in the ${\ensuremath{\sigma}}^{z}$ bases. In the case of the transverse field Ising model, we show that all the ``crystal'' configurations follow the formulas expected from conformal field theory (CFT). In the case of the critical XX chain, we show that the only configurations that follow the formulas of the CFT are the ones which respect the filling factor of the system. By repeating all the calculations in the presence of open and periodic boundary conditions we find further support for our classification of different configurations. Using the developed technique, we also study Shannon information of a subregion in our system. In this respect we distinguish particular configurations that are more important in the study of the scaling limit of the Shannon information of the subsystem. Finally, we study the evolution of formation probabilities, Shannon information, and Shannon mutual information after a quantum quench in free fermion system. In particular, for the initial state considered in this paper, we demonstrate that the Shannon information after quantum quench first increases with the time and then saturates at time ${t}^{*}=\frac{l}{2}$, where $l$ is the size of the subsystem.

arXiv:1511.06401•Physical review. B./Physical review. B•Mar 28, 2016•26 citations

Entanglement and boundary entropy in quantum spin chains with arbitrary direction of the boundary magnetic fields

J. C. Xavier, M. A. Rajabpour

We calculate the entanglement and the universal boundary entropy (BE) in critical quantum spin chains, such as the transverse field Ising chain and the XXZ chain, with arbitrary direction of the boundary magnetic field (ADBMF). We determine the boundary universality class that an ADBMF induces. In particular, we show that the induced boundary conformal field theory depends on the point on the Bloch sphere where the boundary magnetic field directs. We show that the classification of the directions boils down to the simple fact that the boundary field breaks the bulk symmetry or does not. We present a procedure to estimate the universal BE, based on the finite-size corrections of the entanglement entropy, which apply to ADBMF. To calculate the universal BE in the XXZ chain, we use the density matrix renormalization group. The transverse field XY chain with ADBMF after Jordan-Wigner transformation is not a quadratic free fermion Hamiltonian. We map this model to a quadratic free fermion chain by introducing two extra ancillary spins coupled to the main chain at the boundaries, which makes the problem integrable. The eigenstates of the transverse field XY chain can be obtained by proper projection in the enlarged chain. Using this mapping, we are able to calculate the entanglement entropy of the transverse field XY chain using the usual correlation matrix technique up to relatively large sizes.

arXiv:2003.00095•Physical review. B./Physical review. B•Jun 8, 2020•12 citations

Efekti konačne veličine u XY lancu

Vanja Marić

The XY chain in a transverse field is a prototypical exactly solvable model with interesting and desirable properties: its two-dimensional phase diagram, characterized by an anisotropy parameter and the strength of an external magnetic field, hosts two different quantum phase transitions at zero temperature, while the model can always be mapped into a system of free (spin-less) fermions. One of the phase transitions is the universality of free fermions (c = 1 CFT), while the other is that of the 1D quantum Ising model (c = 1=2 CFT). As a matter of fact, the XY chain is a generalization of the Ising chain. While both lines of phase transitions can be described by Conformal Field Theory, the bi-critical point at which these lines meet is non-conformal, since it has a quadratic spectrum. The aim of this thesis is to observe the non-conformal nature of the bi-critical point in the XY chain in a numerical experiment trough quantities accessible also in non-exactly solvable models, such as the entanglement entropy and the eigenvalues of the reduced density matrix. In this way, we want to propose in the future numerical tests whether a multi-critical point in an arbitrary model is conformal or not. First we introduce the XY chain and using standard analytical methods find its microscopic solution. Like the Ising model, the XY chain has a phase of broken Z2 symmetry, where the model shows two degenerate ground states in the thermodynamic limit of infinite system length. Having noticed a scarcity of results on the exact energy degeneracy between two putative ground states, we examine the problem using numerical and analytical methods. The analytical method, employing complex analysis and Fourier series, has given a definite answer for the Ising model. Using this method for the more general XY chain we find the dependence of the exact degeneracy on the number of spins in the special case of zero magnetic field and show the absence of an exact degeneracy when the anisotropy is greater than 1 and the number of spins is even. Coming back to the main aim of the thesis, we derive the reduced density matrix eigenvalues and the entanglement entropy in the XY chain and construct a numerical algorithm for calculating them. We accomplish to discriminate the non-conformal nature of the bi-critical point by comparing the entanglement entropy and the largest reduced density matrix eigenvalue with the conformal field theory predictions near and far from the bi-critical point.

•Jul 14, 2017•0 citations

Global geometric entanglement in transverse-field XY spin chains: finite and infinite systems

Tzu-Chieh Wei, Smitha Vishveshwara, Paul M. Goldbart

The entanglement in quantum XY spin chains of arbitrary length is investigated via the geometric measure of entanglement. The emergence of entanglement is explained intuitively from the perspective of perturbations. The model is solved exactly and the energy spectrum is determined and analyzed in particular for the lowest two levels for both finite and infinite systems. The overlaps for these two levels are calculated analytically for arbitrary number of spins. The entanglement is hence obtained by maximizing over a single parameter. The corresponding ground-state entanglement surface is then determined over the entire phase diagram, and its behavior can be used to delineate the boundaries in the phase diagram. For example, the field-derivative of the entanglement becomes singular along the critical line. The form of the divergence is derived analytically and it turns out to be dictated by the universality class controlling the quantum phase transition. The behavior of the entanglement near criticality can be understood via a scaling hypothesis, analogous to that for free energies. The entanglement density vanishes along the so-called disorder line in the phase diagram, the ground space is doubly degenerate and spanned by two product states. The entanglement for the superposition of the lowest two states is also calculated. The exact value of the entanglement depends on the specific form of superposition. However, in the thermodynamic limit the entanglement density turns out to be independent of the superposition. This proves that the entanglement density is insensitive to whether the ground state is chosen to be the spontaneously $Z_2$ symmetry broken one or not. The finite-size scaling of entanglement at critical points is also investigated from two different view points. First, the maximum in the field-derivative of the entanglement density is computed and fitted to a logarithmic dependence of the system size, thereby deducing the correlation length exponent for the Ising class using only the behavior of entanglement. Second, the entanglement density itself is shown to possess a correction term inversely proportional to the system size, with the coefficient being universal (but with different values for the ground state and the first excited state, respectively).

arXiv:1012.4114•Quantum Information and Computation•Mar 1, 2011•10 citations

Tensor networks, quantum spin chains, and quantum field theory

Yijian Zou

Understanding the universality class of continuous phase transition is of central importance in condensed matter physics. In one spatial dimension, the universal properties are encoded in the conformal field theory (CFT), which is in turn specified by the conformal data. In this thesis, we propose a systematic method to extract complete and accurate conformal data from the critical quantum spin chain based on the operator-state correspondence, thus completing the project initiated by Cardy and others in the 80's. \n \nOur method is based on the low-energy eigenstates of the critical quantum spin chain with periodic boundary conditions. First, scaling dimensions and conformal spins are extracted by the energies and momenta. Second, the primary states and conformal towers are identified by using the Koo-Saleur lattice Virasoro generators. Third, we propose a systematic way of identifying lattice operators with CFT operators, which enables us to compute operator product expansion coefficients from the low-energy eigenstates. Finally, the whole approach is generalized to critical quantum spin chains with antiperiodic boundary condition. In order to reduce finite-size corrections, we put forward the periodic uniform matrix product state (puMPS) algorithm which enables us to compute low-energy eigenstates of a critical quantum spin chain up to several hundreds of spins. Our method also enables us to study nonperturbatively the renormalization group flow between two CFTs as well as explore the emergence of extended symmetries beyond conformal symmetry. \n \nWe test our method with the Ising model and its generalization due to O'Brien and Fendley. The latter model is featured by a tricritical point described by the tricritical Ising CFT and a line of critical points interpolating between the Ising CFT and the tricritical Ising CFT. We extract complete conformal data from the two models and find excellent agreement with analytical results. Furthermore, we study the spectral renormalization group flow between the two CFTs nonperturbatively. At the tricritical point, the underlying CFT has an extended symmetry, the superconformal symmetry. We propose lattice operators that correspond to supervirasoro generators and verify their action on low-energy eigenstates. In this way we study the emergence of superconformal symmetry in critical quantum spin chains.

•Jul 7, 2020•0 citations

Finite-size scaling at quantum transitions

Massimo Campostrini B, Andrea Pelissetto C, Ettore Vicari

We develop the finite-size scaling (FSS) theory at quantum transitions. We consider various boundary conditions, such as open and periodic boundary conditions, and characterize the corrections to the leading FSS behavior. Using renormalization-group (RG) theory, we generalize the classical scaling ansatz to describe FSS in the quantum case, classifying the different sources of scaling corrections. We identify nonanalytic corrections due to irrelevant (bulk and boundary) RG perturbations and analytic contributions due to regular backgrounds and analytic expansions of the nonlinear scaling fields. To check the general predictions, we consider the quantum $XY$ chain in a transverse field. For this model exact or numerically accurate results can be obtained by exploiting its fermionic quadratic representation. We study the FSS of several observables, such as the free energy, the energy differences between low-energy levels, correlation functions of the order parameter, etc., confirming the general predictions in all cases. Moreover, we consider bipartite entanglement entropies, which are characterized by the presence of additional scaling corrections, as predicted by conformal field theory.

arXiv:1401.0788•Physical Review B•Mar 21, 2014•107 citations
Abstract We calculate exactly the probability to find the ground state of the XY chain in a given spin configuration in the transverse σ z -basis. By determining finite-volume corrections to the probabilities for a wide variety of configurations, we obtain the universal boundary entropy at the critical point. The latter is a benchmark of the underlying boundary conformal field theory characterizing each quantum state. To determine the scaling of the probabilities, we prove a theorem that expresses, in a factorized form, the eigenvalues of a sub-matrix of a circulant matrix as functions of the eigenvalues of the original matrix. Finally, the boundary entropies are computed by exploiting a generalization of the Euler–MacLaurin formula to non-differentiable functions. It is shown that, in some cases, the spin configuration can flow to a linear superposition of Cardy states. Our methods and tools are rather generic and can be applied to all the periodic quantum chains which map to free-fermionic Hamiltonians.
We first provide a formula to calculate the probability of occurrence of different configurations (formation probabilities) in a generic free fermion system. We then study the scaling of these probabilities with respect to the size in the case of the critical transverse-field XY chain in the ${\ensuremath{\sigma}}^{z}$ bases. In the case of the transverse field Ising model, we show that all the ``crystal'' configurations follow the formulas expected from conformal field theory (CFT). In the case of the critical XX chain, we show that the only configurations that follow the formulas of the CFT are the ones which respect the filling factor of the system. By repeating all the calculations in the presence of open and periodic boundary conditions we find further support for our classification of different configurations. Using the developed technique, we also study Shannon information of a subregion in our system. In this respect we distinguish particular configurations that are more important in the study of the scaling limit of the Shannon information of the subsystem. Finally, we study the evolution of formation probabilities, Shannon information, and Shannon mutual information after a quantum quench in free fermion system. In particular, for the initial state considered in this paper, we demonstrate that the Shannon information after quantum quench first increases with the time and then saturates at time t∗=l2{t}^{*}=\frac{l}{2}t∗=2l​, where lll is the size of the subsystem.
We calculate the entanglement and the universal boundary entropy (BE) in critical quantum spin chains, such as the transverse field Ising chain and the XXZ chain, with arbitrary direction of the boundary magnetic field (ADBMF). We determine the boundary universality class that an ADBMF induces. In particular, we show that the induced boundary conformal field theory depends on the point on the Bloch sphere where the boundary magnetic field directs. We show that the classification of the directions boils down to the simple fact that the boundary field breaks the bulk symmetry or does not. We present a procedure to estimate the universal BE, based on the finite-size corrections of the entanglement entropy, which apply to ADBMF. To calculate the universal BE in the XXZ chain, we use the density matrix renormalization group. The transverse field XY chain with ADBMF after Jordan-Wigner transformation is not a quadratic free fermion Hamiltonian. We map this model to a quadratic free fermion chain by introducing two extra ancillary spins coupled to the main chain at the boundaries, which makes the problem integrable. The eigenstates of the transverse field XY chain can be obtained by proper projection in the enlarged chain. Using this mapping, we are able to calculate the entanglement entropy of the transverse field XY chain using the usual correlation matrix technique up to relatively large sizes.
The XY chain in a transverse field is a prototypical exactly solvable model with interesting and desirable properties: its two-dimensional phase diagram, characterized by an anisotropy parameter and the strength of an external magnetic field, hosts two different quantum phase transitions at zero temperature, while the model can always be mapped into a system of free (spin-less) fermions. One of the phase transitions is the universality of free fermions (c = 1 CFT), while the other is that of the 1D quantum Ising model (c = 1=2 CFT). As a matter of fact, the XY chain is a generalization of the Ising chain. While both lines of phase transitions can be described by Conformal Field Theory, the bi-critical point at which these lines meet is non-conformal, since it has a quadratic spectrum. The aim of this thesis is to observe the non-conformal nature of the bi-critical point in the XY chain in a numerical experiment trough quantities accessible also in non-exactly solvable models, such as the entanglement entropy and the eigenvalues of the reduced density matrix. In this way, we want to propose in the future numerical tests whether a multi-critical point in an arbitrary model is conformal or not. First we introduce the XY chain and using standard analytical methods find its microscopic solution. Like the Ising model, the XY chain has a phase of broken Z2 symmetry, where the model shows two degenerate ground states in the thermodynamic limit of infinite system length. Having noticed a scarcity of results on the exact energy degeneracy between two putative ground states, we examine the problem using numerical and analytical methods. The analytical method, employing complex analysis and Fourier series, has given a definite answer for the Ising model. Using this method for the more general XY chain we find the dependence of the exact degeneracy on the number of spins in the special case of zero magnetic field and show the absence of an exact degeneracy when the anisotropy is greater than 1 and the number of spins is even. Coming back to the main aim of the thesis, we derive the reduced density matrix eigenvalues and the entanglement entropy in the XY chain and construct a numerical algorithm for calculating them. We accomplish to discriminate the non-conformal nature of the bi-critical point by comparing the entanglement entropy and the largest reduced density matrix eigenvalue with the conformal field theory predictions near and far from the bi-critical point.
The entanglement in quantum XY spin chains of arbitrary length is investigated via the geometric measure of entanglement. The emergence of entanglement is explained intuitively from the perspective of perturbations. The model is solved exactly and the energy spectrum is determined and analyzed in particular for the lowest two levels for both finite and infinite systems. The overlaps for these two levels are calculated analytically for arbitrary number of spins. The entanglement is hence obtained by maximizing over a single parameter. The corresponding ground-state entanglement surface is then determined over the entire phase diagram, and its behavior can be used to delineate the boundaries in the phase diagram. For example, the field-derivative of the entanglement becomes singular along the critical line. The form of the divergence is derived analytically and it turns out to be dictated by the universality class controlling the quantum phase transition. The behavior of the entanglement near criticality can be understood via a scaling hypothesis, analogous to that for free energies. The entanglement density vanishes along the so-called disorder line in the phase diagram, the ground space is doubly degenerate and spanned by two product states. The entanglement for the superposition of the lowest two states is also calculated. The exact value of the entanglement depends on the specific form of superposition. However, in the thermodynamic limit the entanglement density turns out to be independent of the superposition. This proves that the entanglement density is insensitive to whether the ground state is chosen to be the spontaneously Z2Z_2Z2​ symmetry broken one or not. The finite-size scaling of entanglement at critical points is also investigated from two different view points. First, the maximum in the field-derivative of the entanglement density is computed and fitted to a logarithmic dependence of the system size, thereby deducing the correlation length exponent for the Ising class using only the behavior of entanglement. Second, the entanglement density itself is shown to possess a correction term inversely proportional to the system size, with the coefficient being universal (but with different values for the ground state and the first excited state, respectively).
Understanding the universality class of continuous phase transition is of central importance in condensed matter physics. In one spatial dimension, the universal properties are encoded in the conformal field theory (CFT), which is in turn specified by the conformal data. In this thesis, we propose a systematic method to extract complete and accurate conformal data from the critical quantum spin chain based on the operator-state correspondence, thus completing the project initiated by Cardy and others in the 80's. \n \nOur method is based on the low-energy eigenstates of the critical quantum spin chain with periodic boundary conditions. First, scaling dimensions and conformal spins are extracted by the energies and momenta. Second, the primary states and conformal towers are identified by using the Koo-Saleur lattice Virasoro generators. Third, we propose a systematic way of identifying lattice operators with CFT operators, which enables us to compute operator product expansion coefficients from the low-energy eigenstates. Finally, the whole approach is generalized to critical quantum spin chains with antiperiodic boundary condition. In order to reduce finite-size corrections, we put forward the periodic uniform matrix product state (puMPS) algorithm which enables us to compute low-energy eigenstates of a critical quantum spin chain up to several hundreds of spins. Our method also enables us to study nonperturbatively the renormalization group flow between two CFTs as well as explore the emergence of extended symmetries beyond conformal symmetry. \n \nWe test our method with the Ising model and its generalization due to O'Brien and Fendley. The latter model is featured by a tricritical point described by the tricritical Ising CFT and a line of critical points interpolating between the Ising CFT and the tricritical Ising CFT. We extract complete conformal data from the two models and find excellent agreement with analytical results. Furthermore, we study the spectral renormalization group flow between the two CFTs nonperturbatively. At the tricritical point, the underlying CFT has an extended symmetry, the superconformal symmetry. We propose lattice operators that correspond to supervirasoro generators and verify their action on low-energy eigenstates. In this way we study the emergence of superconformal symmetry in critical quantum spin chains.
We develop the finite-size scaling (FSS) theory at quantum transitions. We consider various boundary conditions, such as open and periodic boundary conditions, and characterize the corrections to the leading FSS behavior. Using renormalization-group (RG) theory, we generalize the classical scaling ansatz to describe FSS in the quantum case, classifying the different sources of scaling corrections. We identify nonanalytic corrections due to irrelevant (bulk and boundary) RG perturbations and analytic contributions due to regular backgrounds and analytic expansions of the nonlinear scaling fields. To check the general predictions, we consider the quantum XYXYXY chain in a transverse field. For this model exact or numerically accurate results can be obtained by exploiting its fermionic quadratic representation. We study the FSS of several observables, such as the free energy, the energy differences between low-energy levels, correlation functions of the order parameter, etc., confirming the general predictions in all cases. Moreover, we consider bipartite entanglement entropies, which are characterized by the presence of additional scaling corrections, as predicted by conformal field theory.